Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)

Show that t, e^t, and sin(t) are linearly independent.

Which of the following pair of functions is linearly independent? (а) еЗx, е(х-3) (b) 4, sin?x + cos?x (c) 2sin2x, sinxcosx (d) sinhx, (e* – e*)

Find the upward flux of F = (14e*, -7e, -6) through the first-octant part of the triangle x+2y+3z = 6. Answer: E

the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at (0.9,-0.1).

EX: Find the F.T. of the triangular pulse: g(t) = AA (+) Sols Apply differentiation property of F-T g(t) = 2A S(t+1)_4A S(t) + 2A S(t-I) The F. I for the both sides gives. (jw)² G (w) = 2A (NE_2 +2²) (e j들 (JW)³&(W) = 4A (2 ² + 2² (e 2 G(W) = -4A whe - 8A w²t -₁) 8A ( cos 뜰 -1) = 82 (1-005) 2 sin wt 4 At sin wt 2 Wt 4 2 3/4 N 2A T ZA 0 0 8(1)-A(+) =9c How did he get the value inside the circle Hes -9C2A ہم تم at ot

point If 1 g(u) V5и + 8 then g' (u) =

Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = V3ti + e“j+ 6e"k at t = 0. (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*).) v(0) = a(0) = (Use symbolic notation and fractions where needed. Give your answers in the form (*,*,*). Use t for the parameter that takes all real values.) 1(t) =

Consider a particle moving along a curve so that it is at position f(t) = (2+³, 2t²) at time t. Then the particle is moving parallel to (1, -2) at time t at the point when the direction of motion is =

Determine which of the following pairs of functions are linearly independent. NO_ANSWER f(x) = e5x NO_ANSWER f(t) = et cos(µt) NO_ANSWER cos (30) f(0) = 2 NO_ANSWER f(x) = x³ 9 ◆ 1. g(x) = e5(x-3) 9 9 2. g(t) = et sin(ut),μ‡0 3. g(0) = 20 cos³ (0) – 15 cos(0) - 4. 3 g(x) = |x|³